Hongquan Yu Institute of Mathematical Sciences, Dalian University of Technology, Dalian 6024, China (Submitted September 996-Final Revision December 996). INTRODUCTION Let W(x), fix), g(x) be formal power series with complex coefficients, W(x) * 0, W{0) =, /(0) = g(0) = 0, Then the coefficients {B^n, k), B 2 (n, k)} in the following expansions, W(x)<J(x)f lk\ = B f a k ) x " l n \, (g{x)f /[W(g(x))k\] = ^B 2 (n,k)x n ln\, () ri>k are called a weighted Stirling pair if f(g(x)) = g(f(x)) = x, i.e.,/ g are reciprocal. When W(x) = l, B^in.k) B 2 (n,k) reduce to a Stirling type pair whose properties are exhibited in [7]. In this paper, we shall present a weighted Stirling pair that includes some previous generalizations of Stirling numbers as particular cases. Some related combinatorial arithmetic properties are also discussed. 2. A WEIGHTED STIRLING PAIR Let t, a, P be given complex numbers with a-(*0. Let f(x) = [( + ax)p' a -]//?, g(x) - [(\+px) alp -l]la, W(x) = (l + ax) t/a. Then, in accordance with (), by noting that f(x) g(x) are reciprocal, we have a weighted Stirling pair, denoted by {S(n, k, a, /?; t\ S(n, k, /?, a, -)} = { >, k), B 2 (n, k)}. We call it an (a, /?; /) [resp. a (ft, a; -t)] pair for short. Moreover, one of the parameters a or p may be zero by considering the limit process. For instance, a (,0; 0) [resp. a (0,; 0)] pair is just Stirling numbers of the first second kinds. Note that from the definition of an (a, /?; t) pair the first equation in (), we may obtain the double generating function of S(n, k, a, /?; t) as (l + ax)' /a cxpl (l + ax^a~ l \ = ^S(n,k,a,B-, t)^~u k. (2) If we differentiate both sides of (2) on x, then multiply by (l + ax) compare the coefficients of x n u k, we have S(n, k-l,a,ftt+p) = S(n +, k, a, fi; t) + (na - t)s(n, k, a, /?; t\ (3) if we differentiate both sides of (2) on u then compare the coefficients of x n u k:, we have S(n y k,a,p;t+p) = P(k + l)s(n,k + \a,p;t) + S(n,k,a,P;t). (4) Thus, the recurrence relation satisfied by S(n, k, a, /?; t) may be obtained by combining (3) (4): S(n + \ k, a,p; t) = (t + fik-ari)s(n, k, a,/3; t) + S(n,k-l, a,p, t). (5) 252 [JUNE-JULY
The initial values of S(n, k, a, /?; t) may be verified easily from () because S(n, 0, a, J3; t) = t(t-a)(t-2a) - (t-(n-l)a) for/i>l, S(n,n,a,fi;t) = l for n>0, S(n, k, a, fi;t) = 0 for k>n. Thus, a table of values of S(n, k, a, J3; t) can be given by concrete computations. «" 0 2 3 TABLE. S(n 2t + /}-a (t+fi-2a) +, K, t(t--aa,fi;t) 0 t t(t- -a) t(t- -a) 2 3 3t + 30-3a From (2), we may get the explicit expression for S(n, k, a, J3; t) via the generalized binomial theorem along the lines of (4.) in [6]. For a complex number a, define the generalized factorial of x with increment a by (x\a) n - x(x - a)(x -2a) - - - (x -na + a) for w =,2,..., (x\a) 0 =. Theorem : The (a, /?; t) pair defined by () may also be defined by the following symmetric relations: ((x + /) a) = S(n, K a, ft; t)(x\fi) k ; fc=0 (*!/?) = s(«,*, A «;-0((*+0l«)*- k=0 Proof: The proof of the theorem may be carried out by the same argument used by Howard [6], by showing that the sequences defined by (6) (7) satisfy the same recurrence relations have the same initial values as that of an (a, /?; t) pair. D Examples: Let A, 0 * 0 be two complex parameters. The so-called weighted degenerate Stirling numbers (S x (n, k,a\0),s(n, k, X\0)) were first introduced discussed by Howard [6] with definitions ( l - x ) - ^ " 0 - *>*! =k\y d S l {n,k,x\0)^ V * ) " {\+ex)^n\+exy -\f ^k^si^kx^^-, riik n\ where 9p = \. Now it is clear that (-l)"-^,(»,k, \,X\0) = S{n,k,\,9;9-X) S{n,k,X\6) = S{n,k,6,\;X). The limiting case 0 = 0, X * 0, gives the weighted Stirling numbers (R^n, k, X), R 2 (n, k, X)) discussed by Carlitz ([2], [3]) with definitions (6) (7) (l-xy\-log(\-x)) k = *! *,(», k, X)?- n ' 998] 253
where the weight function et* comes from the limit of ( + 0t) xie as 0 -> 0. It is apparent that ((-l)^%(n, Jfc, X\ R 2 (n, k, X)) forms a (,0; - X) pair. Further examples are the degenerate Stirling numbers [] defined by H - ((i+ftr-i)* = *!2>i,* 0), m where 0// =. It is clear that {(-\) n - k S x {n, k\0\ S(n, k\0)) is a (,0;0) pair. The noncentral Stirling numbers were first introduced by Koutras in [8] with the definitions: (0 = 5> a (»,*X'-«) k ; (t-a)" = ^S a (n,k)(t) k. It is now clear by Theorem that (s a (n, k), S a (n, k)) is a (,0; a) pair. Jc=0 3. REPRESENTATIONS OF WEIGHTED STIRLING PAIRS For r > 0, f r * 0, let F(x) = Z* =r f k x k I k! JF(x) = T% WjX j I j! be two formal power series. Following Howard [6], for complex z, we define the weighted potential polynomial F k (z) by ^l^f" fc=0 '*-*'-* =< F k (zyx*/k\. (8) Moreover, if r >, define the weighted exponential Bell polynomial B n k (0,..., 0, / r, / r+,...) by *P(x)[F(x)]* =.*!fx,(0,..., 0/ r,/ r +,...)*" ln\. (9) The following lemma is due to Howard ([6], Th. 3.). Lemma 2: With F k (z) B nk defined above, we have Now, from (9) with W(x) = ( + ox:)' 7 " F(x) = [( + ax) /3,a -] / /?, we have S{n 9 k,a 9 frt) = B^k(\p-a 9 tf-a)<fi-2a\w^ (0) Define the weighted potential polynomials A k (z) by 254 [JUNE-JULY
If we differentiate both sides of () with respect to x, then multiply by l + ax compare the coefficients of x k, we obtain It follows that with initial conditions za k {z +) = (z - k)a k {z) + k(t + (a - fi)z -(k- \)d)a k _ x {z). (-l) k [ k ~r i y k (rr + l) = (-d k { k ~ k "y k (n)ht + (a-fi) n -(k-vax-lf-^-^yu"), ^~" 0 " J4 ) («+ l) = l,for»>0, (3) (-!)"(~^\A n (n + \) = (t + a-p){t + a-2p)---{t + a-np), for n>\. (4) Therefore, by equations (2)-(4), the recurrence relations satisfied by S(n,n-k,fi, a; t + a- fl) [may be deduced from (5)] its initial values, we have that {_ l f(k-n-^ak{n + \) = S{n,n-k,p,a-t + a-p). It then follows from Lemma 2, by taking r = (0) that (2) By symmetry, we have the following representation formulas for weighted Stirling pairs. Theorem 3: For S(n 9 k 9 a, /?; t) defined by () S(n, k,f3,a;t + a-{t) defined in a like way, we have (»,*, a, 0 = l V l ) { ^ J ^ ( 5 > S(n, k,fi,cr,t + a-fl = %(-&$-k*}){ ^ ( 6 > Remark: It should be pointed out that similar representation results for the particular case when a = 0, 0=, t = l-x has been proved by Howard [6]. Here we borrow his proof techniques. 4. CONGRUENCE PROPERTIES OF WEIGHTED STIRLING PAIRS A formal power series $(x) = E w > 0 ^ " /«! is called a Hurwitz series if all of its coefficients are integers. It is well known that, for the Hurwitz series flx) with a Q = 0, the series {(f>{x)) k lk\ is again a Hurwitz series for any positive integer k. 998] 255
In this section we always assume a,/3,t e Z. Then it is clear that both (f(x)) k /k\ (g(x)) k /k\ in () are Hurwitz series, so that S(n, k, a, J3; t) S(n, k, /?, a; -i) are two integer sequences. First, let t = 0. Then we have Theorem 4: Let p be a prime number let k y be integers such that j + < k < p. Then the following congruence relation holds: S(p + j,k,p,a;0)^0(modp). (7) Proof: Assume first that a 4 0 (mod /?). For a polynomial ^(x) of degree n in x, we may express it, using Newton's interpolation formula, in the form <f>(x) = <f>(a 0 ) + k=\ y [a 0 a l,..a k ]{x\a} k, where [a 0 a l...a k ] denotes the divided difference at the distinct points x = a Q, a l?..., a k9... {x a) k =(x- a 0 )(x - a x ) (x - a k _ x ). Moreover, we have [a 0 a l...a k ] = a 0 a x CLu a\ l <f>(a Q ) a k ~ l Mad <4 <l>(a k ) a 0 a x CCu k-i o k- «0 of a * a> Now take (/> p {x) = (x\fi) p, then ^,(0) = 0. We have, by (7) (8), that (8) (9) 5(p,*,/?,a;0) = [a 0 a...aj, (20) which may be expressed as a quotient of two determinants as in (9), where a^ - ja (j = 0,,2,...). Notice that the classical argument of Lagrange that applied to the proof of may also be applied to prove the relation (x-l)->(x-p + l) = x p - l -l (mod/?) <f> p (x) = (x\p) p = x(x-fl)...( x -(p-i)0) S x'-fir-ix (mod/7), (2) where the congruence relation between polynomials are defined as usual (cf. [4], pp. 86-87, Th. 2). Also, using Fermat's Little Theorem, we find WaXJaY-fi^Ua)* \ja (mod/?), if/? /?, 0 (mod/?), ifp\p, where j = 0,,2,... Consequently, we obtain, witha^ = ja for k >, Moreover, the denominator is given by a 0 a k ~ l j p (i a x a\~ l <j> p {a x )\ = 0 (mod/?). a k - <A~ l t P (<* k ) 256 [JUNE-JULY
a la ka a k {kaf~ l a k {kaf = a' Y\(j-/) # 0 for k < p (mod/?). 0<i<j<k Thus, we have that S(p, k, fi, oc; 0) = 0 (mod /?) for < k < p. Furthermore, let F(x) = (x \fi) p+j. We then have F(x) =! / S(p+j, k, fi, a; 0)(x a) k F(x) = 0 p (x)(x-pfi)..-(x-(p + j)fi+fi) ^ (xp - pp- l x)x(x - fi)... (x - (J - \)fi) (mod /?) = (x p - fi p ~ l x){x j + ap s - x + + a^x) (mod /?), (22) where a l9..., a j-x e Z. Consequently, we have, for < i < p + y, F(ia) = (mod/>), if Pi A (ia) y+ + ^(/a) 7 + + aj^qaf (mod/?), if/?\fi. Since j < k -, we have a 0 aj,fc-l 0 F(a 0 ) F(a,) = 0 (mod/?), a t a,k-i where the last column is a linear combination of the first k columns modulo/?. Again, the same denominator determinant is not congruent to zero modulo /? for k <p. Thus, we have that S(p + j, k, fi, a; 0) = 0 (mod/?) for j + < k < p. The case for a = 0 (mod/?) may be proved directly using (7), (2), (22) by comparing the corresponding coefficients of powers of x in both sides of (2) (22). Hence, the theorem is proved. Note that in the particular case in which a =, fi = 0 or fi=, a = 0, Theorem 4 reduces to congruences for Stirling numbers of the first second kinds; see [5] for other congruences for Stirling numbers. Corollary 5: Let a, fi, t be integers. Then the (a, fi; t) pair satisfies the basic congruence S(p, k, a, fi;t) = 0 (mod p), (23) where/? is a prime \<k<p. Proof: Let W(x) = (l + ax) t/a = I dn > 0 a rj x n /n\ with a n ez,a 0 = l. Then it is clear from () that J]S(n,k,a,fi;t)x»M = \^a n xvn\ \Ws(n,k,a,fi;0)x«ln\ «>o so that we have S(p, k, a, fi; t) = f.a^sq, k, a, fi; 0){ p \ i=k V / 998] 257
From Theorem 4 (taking j = 0) the fact that (?) = 0 (mod p) for 0 < i < p, it follows that S(p, k, a, /?; r) = 0 (mod/?), the corollary is proved. ACKNOWLEDGMENT The author is very grateful to the referee for his valuable suggestions which considerably improved the presentation of this article. REFERENCES. L. Carlitz. "Degenerate Stirling, Bernoulli Eulerian Numbers." UtilitasMath. 5 (979): 5-88. 2. L. Carlitz. "Weighted Stirling Numbers of the First Second Kind-L" The Fibonacci Quarterly 8.2 (980): 47-62. 3. L. Carlitz. "Weighted Stirling Numbers of the First Second Kind-II." The Fibonacci Quarterly 8.3 (980):242-57. 4. G. H. Hardy & E. M. Wright. An Introduction to the Theory of Numbers. Oxford: Oxford University Press, 98. 5. F. T. Howard. "Congruences for the Stirling Numbers Associated Stirling Numbers." Acta Arithmetica 55 (990):29-4. 6. F. T. Howard. "Degenerate Weighted Stirling Numbers." Discrete Math. 57 (985):45-58. 7. L. C. Hsu. "Some Theorems on Stirling-Type Pairs." Proceedings of Edinburgh Math. Soc. 36 (993):525-35. 8. M. Koutras. "Non-Central Stirling Numbers Some Applications." Discrete Math. 42 (982):73-89. AMS Classification Numbers: 05A5, B73, A07 NEW EDITOR AND SUBMISSION OF ARTICLES Starting March,998 9 all new articles must be submitted to: PROFESSOR CURTIS COOPER Department of Mathematics Computer Science Central Missouri State University Warrensburg, MO 64093-5045 e-mail:' cnc885@cmsu2xmsu.edu Any article that does not satisfy all of the criteria as listed on the inside front cover of the journal will be returned immediately. [JUNE-JULY